Abstract
Public transportation networks are typically operated with a periodic timetable. The periodic event scheduling problem (PESP) is the standard mathematical modeling tool for periodic timetabling. PESP is a computationally very challenging problem: For example, solving the instances of the benchmarking library PESPlib to optimality seems out of reach. Since PESP can be solved in linear time on trees, and the treewidth is a rather small graph parameter in the networks of the PESPlib, it is a natural question to ask whether there are polynomial-time algorithms for input networks of bounded treewidth, or even better, fixed-parameter tractable algorithms. We show that deciding the feasibility of a PESP instance is NP-hard even when the treewidth is 2, the branchwidth is 2, or the carvingwidth is 3. Analogous results hold for the optimization of reduced PESP instances, where the feasibility problem is trivial. Moreover, we show W[1]-hardness of the general feasibility problem with respect to treewidth, which means that we can most likely only accomplish pseudo-polynomial-time algorithms on input networks with bounded tree- or branchwidth. We present two such algorithms based on dynamic programming. We further analyze the parameterized complexity of PESP with bounded cyclomatic number, diameter, or vertex cover number. For event-activity networks with a special—but standard—structure, we give explicit and sharp bounds on the branchwidth in terms of the maximum degree and the carvingwidth of an underlying line network. Finally, we investigate several parameters on the smallest instance of the benchmarking library PESPlib.
Highlights
Creating and optimizing timetables is substantial for planning and operating public transportation networks
We prove that the feasibility version of periodic event scheduling problem (PESP) is fixed-parameter tractable when parameterized by the cyclomatic number, i.e., the dimension of the cycle space of the event-activity network
As PESP instances arising from public transportation networks typically have a special structure, we show in Sect. 6 that on this type of event-activity networks, the branchwidth can be related to invariants of an underlying line network: Roughly speaking, the number of lines at a station of a public transport network is a lower bound on the branchwidth of the event-activity network
Summary
Creating and optimizing timetables is substantial for planning and operating public transportation networks. There is a function f such that for Journal of Scheduling a given graph G on n vertices with a nice tree decomposition of treewidth ≤ k and a natural number T , there is an O( f (k, T ) · n) algorithm deciding the T -Vertex Coloring problem on G (Arnborg & Proskurowski, 1989) This motivates the investigation of parameterized complexity of PESP. The existence of fixed-parameter tractable algorithms allowing a unary encoding of the period time T is shown to be unlikely, as we show that deciding the feasibility of a PESP instance is W[1]-hard w.r.t. the vertex cover number and for the treewidth.
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